Question: A composite function can be written as $w\bigl(u(x)\bigr)$, where $u$ and $w$ are basic functions. Is $h(x)=\cos^9(x)$ a composite function? If so, what are the "inner" and "outer" functions? Choose 1 answer: Choose 1 answer: (Choice A) A $h$ is composite. The "inner" function is $\cos(x)$ and the "outer" function is $x^9$. (Choice B) B $h$ is composite. The "inner" function is $x^9$ and the "outer" function is $\cos(x)$. (Choice C) C $h$ is not a composite function.
Explanation: Composite and combined functions A composite function is where we make the output from one function, in this case $u$, the input for another function, in this case $w$. We can also combine functions using arithmetic operations, but such a combination is not considered a composite function. The inner function The inner function is the part we evaluate first. Frequently, we can identify the correct expression because it will appear within a grouping symbol one or more times in our composed function. Here, we only have $x$ inside of the parentheses, which is trivial. However, we can rewrite our expression as $(\cos(x))^9$. Now we have $\cos(x)$ inside a grouping symbol. We evaluate this expression first, so $u(x)=\cos(x)$ is the inner function. The outer function Then we raise the entire output of $u$ to the power of $9$. So $w(x)=x^9$ is the outer function. Answer $h$ is composite. The "inner" function is $\cos(x)$ and the "outer" function is $x^9$. Note that there are other valid ways to decompose $h$, especially into more complicated functions.